{
"info": {
"author": "Eugene Prilepin",
"author_email": "esp.home@gmail.com",
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"classifiers": [
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"Intended Audience :: Developers",
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"License :: OSI Approved :: MIT License",
"Programming Language :: Python",
"Programming Language :: Python :: 3",
"Programming Language :: Python :: 3.5",
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"Programming Language :: Python :: 3.7",
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"Topic :: Scientific/Engineering",
"Topic :: Scientific/Engineering :: Mathematics",
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"description": "# CSAPS: Cubic spline approximation (smoothing)\n\n[](https://pypi.python.org/pypi/csaps)\n[](https://travis-ci.org/espdev/csaps)\n\n[](LICENSE)\n\nThis module provides cubic smoothing spline for univariate/multivariate/gridded data approximation.\nThe smoothing parameter can be calculated automatically or it can be set manually. \n\nThe smoothing parameter should be in range `[0, 1]` where bounds are:\n* 0: The smoothing spline is the least-squares straight line fit to the data\n* 1: The natural cubic spline interpolant\n\nThe calculation of the smoothing spline requires the solution of a linear system \nwhose coefficient matrix has the form `p*A + (1 - p)*B`, with the matrices `A` and `B` \ndepending on the data sites `X`. The automatically computed smoothing parameter makes \n`p*trace(A) equal (1 - p)*trace(B)`.\n\n## Installation\n\nPython 3.5 or above is supported.\n\n```\npip install csaps\n```\n\nThe module depends only on NumPy and SciPy.\n\nOn Windows we highly recommend to use unofficial builds [NumPy+MKL](https://www.lfd.uci.edu/~gohlke/pythonlibs/#numpy) \nand [SciPy](https://www.lfd.uci.edu/~gohlke/pythonlibs/#scipy) from Christoph Gohlke.\n\n```\n> mkdir depends\n\ndownload numpy (https://www.lfd.uci.edu/~gohlke/pythonlibs/#numpy) to \"depends\" directory\ndownload scipy (https://www.lfd.uci.edu/~gohlke/pythonlibs/#scipy) to \"depends\" directory\n\n> pip install --find-links depends/ csaps\n```\n\n## Smoothing univariate data\n\nUnivariate data are two vectors: X and Y with the same size. X is data sites, Y is data values.\nFor univariate case X-values must satisfy the condition: `x1 < x2 < ... < xN`.\n\nYou can use `UnivariateCubicSmoothingSpline` class for uivariate data smoothing.\n\n```python\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport csaps\n\nnp.random.seed(1234)\n\nx = np.linspace(-5., 5., 25)\ny = np.exp(-(x/2.5)**2) + (np.random.rand(25) - 0.2) * 0.3\n\nsp = csaps.UnivariateCubicSmoothingSpline(x, y, smooth=0.85)\n\nxs = np.linspace(x[0], x[-1], 150)\nys = sp(xs)\n\nplt.plot(x, y, 'o', xs, ys, '-')\nplt.show()\n```\n\n\n\n### Smoothing weighted univariate data\n\nThe algorithm supports weighting. You can set weights vector that will determine \nweights for all data values:\n\n```python\nimport csaps\n\nx = [1., 2., 4., 6.]\ny = [2., 4., 5., 7.]\nw = [0.5, 1, 0.7, 1.2]\n\nsp = csaps.UnivariateCubicSmoothingSpline(x, y, w)\n...\n```\n\n### Smoothing univariate data with vectorization\n\nThe algorithm supports vectorization. You can compute smoothing splines for \n`X`, `Y` data where `X` is data site vector and `Y` is ND-array of data value vectors. \nThe shape of `Y` array must be: `(d0, d1, ..., dN)` where `dN` must equal of `X` vector size.\n\nIn this case the smoothing splines will be computed for all `Y` data vectors at a time.\n\nFor example:\n\n```python\nimport numpy as np\nimport csaps\n\n# data sites\nx = [1, 2, 3, 4]\n\n# two data vectors\ny = np.array([(2, 4, 6, 8), \n (1, 3, 5, 7)])\n\nsp = csaps.UnivariateCubicSmoothingSpline(x, y)\n\nxi = np.linspace(1, 4, 10)\nyi = sp(xi)\n\nprint(yi.shape) # (2, 10)\nassert yi.shape[:-1] == y.shape[:-1]\nassert yi.shape[-1] == xi.size\n```\n\n**Important**:\nThe same weights vector and the same smoothing parameter will be used for all Y data.\n\n## Smoothing multivariate data\n\nWe can easily smooth multivariate data using univariate smoothing spline, vectorization and parametrization.\nWe should define parametric data sites vector `t` with condition `t1 < t2 < ... < tN` and make spline for each dimension `X(t), Y(t), ..., M(t)`.\n\nThe module provides `MultivariateCubicSmoothingSpline` class for this case.\n\nThis class is a simple wrapper and it just automatically computes `t` vector by default and calls `UnivariateCubicSmoothingSpline`:\n\n```python\n# Construct multivariate spline from t and X, Y, Z, ..., M\nsx = UnivariateCubicSmoothingSpline(t, data[x])\nsy = UnivariateCubicSmoothingSpline(t, data[y])\nsz = UnivariateCubicSmoothingSpline(t, data[z])\n...\nsm = UnivariateCubicSmoothingSpline(t, data[m])\n\n# Or the same with using vectorization\nsxyz_m = UnivariateCubicSmoothingSpline(t, data)\n```\n\n### Smoothing 3D noisy parametric curve example\n\nIn this example we make parametric 3D curve from `theta` parameter and compute spline from the same `theta` parameter.\nIn fact we could use `UnivariateCubicSmoothingSpline` with vectorization instead of `MultivariateCubicSmoothingSpline` in this case.\n\n```python\nimport numpy as np\n\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\n\nimport csaps\n\nn = 100\ntheta = np.linspace(-4 * np.pi, 4 * np.pi, n)\nz = np.linspace(-2, 2, n)\nr = z ** 2 + 1\nnp.random.seed(1234)\nx = r * np.sin(theta) + np.random.randn(n) * 0.3\nnp.random.seed(5678)\ny = r * np.cos(theta) + np.random.randn(n) * 0.3\n\ndata = np.vstack((x, y, z))\n\nsp_theta = csaps.MultivariateCubicSmoothingSpline(data, theta, smooth=0.95)\n\n# or the same\n# sp_theta = csaps.UnivariateCubicSmoothingSpline(theta, data, smooth=0.95)\n\ntheta_i = np.linspace(-4 * np.pi, 4 * np.pi, 250)\ndata_i = sp_theta(theta_i)\n\nxi = data_i[0, :]\nyi = data_i[1, :]\nzi = data_i[2, :]\n\nfig = plt.figure()\nax = fig.gca(projection='3d')\nplt.show()\n```\n\n![\"2019-04-04_02-28-52\"](\"https://user-images.githubusercontent.com/1299189/55519724-a75a6580-5681-11e9-8beb-2111664f8ee3.png\")
\n\n## Smoothing ND-gridded data\n\nThe algorithm can make smoothing splines for ND-gridded data approximation.\nIn this case we use coordinatewise smoothing (tensor-product of univariate splines coefficients).\n\nYou can use `NdGridCubicSmoothingSpline` class for ND-gridded data smoothing. \nYou also can set weights and smoothing parameters for each dimension.\n\nCurrently the implementation does not support vectorization for ND-gridded data.\n\n### Smoothing 2D-gridded data (surface) example\n\n```python\nimport numpy as np\n\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\n\nimport csaps\n\nxdata = [np.linspace(-3, 3, 61), np.linspace(-3.5, 3.5, 51)]\ni, j = np.meshgrid(*xdata, indexing='ij')\n\nydata = (3 * (1 - j)**2. * np.exp(-(j**2) - (i + 1)**2)\n - 10 * (j / 5 - j**3 - i**5) * np.exp(-j**2 - i**2)\n - 1 / 3 * np.exp(-(j + 1)**2 - i**2))\n\nnp.random.seed(12345)\nnoisy = ydata + (np.random.randn(*ydata.shape) * 0.75)\n\nsp = csaps.NdGridCubicSmoothingSpline(xdata, noisy, smooth=0.988)\nysmth = sp(xdata)\n\nfig = plt.figure()\nax = fig.add_subplot(111, projection='3d')\n\nax.plot_wireframe(j, i, noisy, linewidths=0.5, color='r')\nax.scatter(j, i, noisy, s=5, c='r')\n\nax.plot_surface(j, i, ysmth, linewidth=0, alpha=1.0)\n\nplt.show()\n```\n\n![\"2019-03-22_10-22-59\"](\"https://user-images.githubusercontent.com/1299189/54817564-2ff30200-4c8f-11e9-8afd-9055efcd6ea0.png\")
\n\n## More examples\n\nPlease look through [csaps.ipynb](examples/csaps.ipynb) file for more examples.\n\n## Testing\n\nWe use pytest and tox (on Travis CI) for testing.\nPlease see [test_csaps.py](tests/test_csaps.py) file.\n\n## Algorithms and implementations\n\n`csaps` is a Python modified port of MATLAB [CSAPS](https://www.mathworks.com/help/curvefit/csaps.html) function that is an implementation of \nFortran routine SMOOTH from [PGS](http://pages.cs.wisc.edu/~deboor/pgs/) (originally written by Carl de Boor).\n\n[csaps-cpp](https://github.com/espdev/csaps-cpp) C++11 Eigen based implementation of the algorithm.\n\n## References\n\nC. de Boor, A Practical Guide to Splines, Springer-Verlag, 1978.\n\n\n",
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